Modular Form Data by Ken-ichi SHIOTA


Summary

I have calculated the follwing data on the spaces M20(q)) of the elliptic modular forms of weight 2, for all prime levels 11 <= q < 10000:
  • The (q,∞)-quaternion algebra D over Q
  • A maximal order O in D
  • The class number H and the type number T of D
    • H is equal to the dimension of M20(q)).
    • T is equal to the dimension of the minus-eigenspace M2-0(q)) of the Atkin-Lehner involution.
  • The representatives { Aj } j = 1,...,H of the left O-ideal classes
  • The maximal orders Oj = (Aj)-1Aj in D
  • The representatives { Aij } i = 1,...,H of the left Oj-ideal classes
    • Aij is obtained by Aij = ( a constant multiple of ) (Aj)-1Ai.
  • The group order ej of the unit group of Oj
  • The theta series θij associated to the norm form of Aij
    • The definition of the theta series is
           θij(z) = (1/ej) Σ x in Aij exp(2πiz N(x)/N(Aij))
      where N denotes the reduced norm on D.
    • We have ej θij = ei θji, hence we have to calculate θij only for i >= j.
  • The Brandt matrices B(n) ( n = 0,1,2,... )
    • The definition of the Brandt matrices is
           Σ n = 0,1,2,... B(n) exp(2πinz) = ( θij(z) ) i,j = 1,...,H
    • For n >= 1, B(n) is an integral representation matrix of the Hecke operator T(n) on M20(q)).
  • The characteristic polynomial of B(n) ( = that of T(n) ) and its factorization over Z
Further, I am calculating the follwing data:
  • The common eigenvectors of B(n)'s, and the eigenvalues of B(n0) for a selected n0
    • Each eigenvector corresponds to the Eisenstein series E or a newform f in the space of the elliptic cusp forms S20(q)).
    • Each eigenvalue is equal to a Hecke eigenvalue, hence to a Fourier coefficient of E or f.

Reference

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